efficient cva computation by risk factor decomposition

Efficient CVA computation by risk factor decomposition

Kees de Graaf, Drona Kandhai & Christoph Reisinger

CSL Colloquium Amsterdam, January 2016

de Graaf (2016) Efficient CVA December 2015 1 / 33

1 Credit Valuation Adjustment (CVA) and Counterparty Credit Risk(CCR)

2 High dimensional problems

3 The forward Kolmogorov PDE

4 Dimension reduction

5 Results


Example - I

Suppose you bet with a friend on a Monday in Amsterdam on a day inOctober

You will get 100 euro from your friend if :it rains every single day of the comming week

You’re friend gets 100 euro from you if:it it is dry for at least one day of the comming week

After one week you exchange the money


Example - II

Suppose it rained every day, but:

Your friend gets his bank account hacked with all the money gone;

Your friend spends all his money during the week;

Your friend disappears to Ibiza...

Whom should you charge for the earned, but lost 100 euro? Too late tothink about it after it happened.


Intuition of Counterparty Credit Risk (CCR)

CCR analyses risks arising not from the choice of what and where togamble, but with whom.

Exposure measures the money at risk depending on what yougamble.

Credit Value Adjustment (CVA) is a price you charge (or pay) inorder to insure against a possible counterparty default.

Quantiles present worst/best case scenarios.


CVA definition

According to Basel III:

Credit Value Adjustment (CVA) ”is the difference between therisk-free portfolio value and the true portfolio value that takes intoaccount the possibility of a counterparty’s default”


CVA Formula

Mathematically:

CVA(t,T ) = (1− δ)

∫ T

tEQ [PE (s)|τ = s] dPD(s) (1)

In practice:

CVA(t,T ) ≈ (1− δ)N∑

k=1

q(tk−1, tk)EPE (tk) (2)

PE (t) = Positive ExposureQ = Risk neutral measureδ = recovery rateτ = default timeEPE (t) = (discounted) Expected Positive ExposurePD(t) = probability density of default before tq(tk−1, tk) = default prob. in (tk−1, tk)


Recovery Rate and default probability

Two important ingredients of CVA:

I. Recovery rate δ:In practice can be deduced from CDSTaken constant

II. Probability of default PD(t):Needed at every time t ∈ [t0,T ]Can be modeled or also taken from CDS quotes


Exposure

We will focus on the main ingredient of CVA:

III. Expected Positive Exposure EPE (t):Positive (discounted) future valueCan be computed with Monte Carlo or Forward Kolmogorov PDE

Note that there can be a correlation between default probability andexposure (Wrong Way Risk)

(A lot of rain ⇐⇒ friend in Ibiza)


Earlier related work

Computing Exposure:

[Ng and Peterson (2009)] and [Ng et al. (2010)]: Longstaff-Schwarztechnique compared to FD and nested MC

[de Graaf et al. (2015)]: FDMC for Exposure of portfolios andsensitivities

[Simaitis et al. (2015)]: Impact of stochastic volatility and rates in CCR

Dimension reduction:

[Reisinger and Wissman (2015a)] and[Reisinger and Wissman (2015b)]: Methodology and accuracy of lowerdimensional approximations of high-dimensional PDEs


High dimensionality

CVA is especially important for portfolios

Due to correlation, risk factors cannot be modeled independently

Finite Difference method cannot handle dimensions greater than 4(curse of dimensionality)

That is why Monte Carlo is industry standard (scalable)


Disadvantages of Monte Carlo

Can also be computationally heavy

Can be problematic for fat tails (Heston and the Feller condition)

Simulation based (non-deterministic)

Can be problematic in the tails of the distribution


Challenge

Compute risk measures for a portfolio of FX derivatives assuming manyrisk factors, based on real data.

a. FX derivatives:Typically traded in portfoliosCross Currency Swaps

b. many risk factors:Multiple currenciesSmileStochastic interest rates

c. real data:Calibrated models


Model Choices

With many risk factors we mean, extend the Black-Scholes model toinclude:

a. Stochastic Volatility:Skew and Smile in FX marketModels: Heston, SABR

b. Stochastic rates:Vital for long dated derivativesHull-White model includes the term structure

Note that this leads to high dimensional models


General underlying dynamics

For just one FX rate St , this comes down to:

H-2HW SDEs:

dSt = (rdt − r ft )Stdt +

√VtStdW

1t ,

dVt = κ(η − Vt)dt + σ√VtdW

2t ,

drdt = λd(θd(t)− rdt )dt + ηddW 3t ,

dr ft =[λf (θf (t)− r ft ) + ηf ρ1,4

√Vt

]dt + ηf dW 4

t ,

dW it dW

jt = ρi ,jdt, for i 6= j ∈ 1, . . . 4


Resulting PDE

4 dimensional (Kolmogorov-forward) probability density P of one single FXrate:

∂P

∂t− 1

2

∂2

∂s2

(s2vP

)− 1

2

∂2

∂v2

(γ2vP

)− 1

2

∂2

∂(rd)2

(η2dP)− 1

2

∂2

∂(r f )2

(η2f P)

+∂

∂s

((rdτ − r fτ )sP

)+

∂

∂v(κ(v̄ − v)P) +

∂

∂rd

(λd(θd(T − τ)− rdτ )P

)+

∂

∂r f

(λf (θf (T − τ)− r fτ − ρS ,r f ηf

√v)P

)+ ρS,v

∂2

∂s∂v(γsvP)

+ ρS ,rd∂2

∂s∂rd(ηds√vP)

+ ρS ,r f∂2

∂s∂r f(ηf s√vP)

+ . . . = 0

for three FX rates this will be a 10-dimensional PDE...


Solving the Kolmogorov-Forward PDE

We discretize in every dimension

Partial space derivatives ⇒ finite differences

Use Alternating Direction Implicit scheme for time stepping (see[in ’t Hout and Foulon(2010)] )

Adjusted for the forward Kolmogorov (see [Itkin. (2015)] )


Combining density and payoff

In one plot the procedure is summarized as follows:

0 100 200 300 400 500 60025

30

35

40

45

50

Vt

time

t = t0 t = tm

at every (Xmi ),

calculate

V (Xmi , tm)

and compute EE as:∑N

i=1 P (Xmi , tm)V (Xm

i , tm)

Density

on a grid

t = T


Portfolio of Cross Currency Swaps

We are looking at fixed notional floating vs floating cross currencybasis swaps

Notional is exchanged at inception and maturity

The swaps are traded ATM (all cash flows are converted in domesticcurrency by the spot FX rate)

EPE is given by:

EPE (t) = max (0,VEURUSD(t) + VEURGBP(t) + VEURJPY(t))


Dimension Reduction - I

I. Lower dimensional approximations:

First approximate V assuming only X 1t is stochastic:

V (X 1t ) := V (X 1

t , fX 1 (t), . . . , fX 10 (t)).

where you define fX i s.t.:

fX i (t) = E[X it |X i

t0= X i

0

]then, compute the higher dimensional approximations:

V (X 1t ,X

2t ) :=V (X 1

t ,X2t , fX 3 (t), . . . , fX 10 (t)),

V (X 1t ,X

3t ) :=V (X 1

t , fX 2 (t),X 3t , . . . , fX 10 (t)),

...

V (X 1t ,X

10t ) :=V (X 1

t , fX 2 (t), . . . , fX 9 (t),X 10t ).


Dimension Reduction - II

II. Truncating:

For CCY: we choose only the FX rates as a base

For three dimensions, for example:

V (X 1t ,X

2t ,X

3t ) = V (X 1

t ) + (V (X 1t ,X

2t )− V (X 1

t ))

+ (V (X 1t ,X

3t )− V (X 1

t ))

+(V (X 1

t ,X2t ,X

3t )− V (X 1

t ,X2t )

− V (X 1t ,X

3t )− V (X 1

t ))

We hope that the red part is not so big...

In higher dimensions, corrections will also be higher dimensional


Dimension Reduction - III

Definition

Let V (X 1t , . . . ,X

nt ) be the value of a portfolio, driven by n risk factors,

than the k-th 2d ANOVA approximation (where k ∈ [1, . . . , d ]) equals:

V kA(X 1

t , . . . ,Xnt ) := V (X k

t ) +∑n

i 6=k V (X kt )− V (X k

t ,Xit ) (3)

= (2− d)V (X kt ) +

∑ni 6=k V (X k

t ,Xit ), (4)

Decomposing the problem

Note that here only one and two-dimensional corrections are needed

We can also use higher order corrections


How to find our ”best” ANOVA approximation?

Use the different ANOVA approximations as control variates andminimize:

mink∈(1,2,...,n)

Var(V kMC(X 1

t , . . . ,Xnt )), (5)

where V kMC(X 1

t , . . . ,Xnt ) defines the Monte Carlo estimator of

V (X 1t , . . . ,X

nt ) which uses V k

A(X 1t , . . . ,X

nt ) as a control variate

Only a few sample paths are sufficient


Market Volatilities

Comparing two used volatility surfaces for EURUSD and EURJPY FXrate, two completely different markets

0.8 1 1.2 1.4 1.6 1.8 2 9%

10%

11%

12%

13%

14%

15%2008 EURUSD

Moneyness

Volatility

6M1Y2Y5Y10Y

(b) Volatility surface in EURUSD.

0.8 1 1.2 1.4 1.6 1.8 2 2.2 8%

10%

12%

14%

16%

18%

20%

22%2008 EURJPY

Moneyness

Volatility

6M1Y2Y5Y10Y

(c) Volatility surface in EURJPY.

Smile is inverted (increasing vs decreasing term structure).

Skew is more present in EURJPY.


Expected Exposure

Two EE profiles approximated with the help of different PCAs

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−14

−12

−10

−8

−6

−4

−2

0

2

4

62008 PCA: EURUSD

Time

Exposure

u(f 1)u(f 1, v1)u(f 1, f 2)u(f 1, f 3)u(f 1, rd)u(f 1, rf )UPCA (f

1)

UMC (~θ)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−14

−12

−10

−8

−6

−4

−2

0

2

4

62008 PCA: EURJPY

Time

Exposure


3)

UMC (~θ)

If we look at relative difference, the EURUSD rate seems to beslightly more important

Note that the Notional of the EURJPY CCYS is less than thenotional of the EURUSD CCYS


Expected Positive Exposure

Two EPE profiles approximated with the help of different PCAs

0 1 2 3 4 50

5

10

152008 PCA: EURUSD

Time

PositiveExposure


1)

UMC (~θ)

0 1 2 3 4 50

5

10

152008 PCA: EURJPY

Time

PositiveExposure


3)

UMC (~θ)

Again the EURUSD rate seems to be more important


3 dimensional corrections - I

If we add three dimensional corrections, results get better

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−14

−12

−10

−8

−6

−4

−2

0

2

4

62008 PCA: EURUSD

Time

Exposure

u(f 1)u(f 1, v1)u(f 1, f 2)u(f 1, f 3)u(f 1, rd)u(f 1, rf )u(f 1, f 2, v2)u(f 1, f 3, v3)UPCA (f

1)

UMC (~θ)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−14

−12

−10

−8

−6

−4

−2

0

2

4

62008 PCA: EURJPY

Time

Exposure


3)

UMC (~θ)

Both EURUSD and EURJPY improve significantly


3 dimensional corrections - II

The same holds for EPE:

0 1 2 3 4 50

5

10

152008 PCA: EURUSD

Time

PositiveExposure


1)

UMC (~θ)

0 1 2 3 4 50

5

10

152008 PCA: EURJPY

Time

PositiveExposure


3)

UMC (~θ)

Downside is that we need to solve 3d PDEs (but we can do that!)


Variance reduction Expected Positive Exposure - I

The Variance reduction confirms our findings

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.452008 EE PCA Variance reduction

Time

PCA

var/va

r

CV: EURUSD PCACV: EURGBP PCACV: EURJPY PCA

(l) 2d corrections

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Time

PCA

var/va

r


(m) 3d corrections

For EE the EURUSD rate is the best PCA

Dimension reduction of around 5 for 2d corrections and almost 100for 3d corrections


Variance reduction Expected Positive Exposure - II

Similar for EPE

0 1 2 3 4 50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.552008 EPE PCA Variance reduction

Time

PCA

var/va

r


(n) 2d corrections

0 1 2 3 4 50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.552008 EPE PCA Variance reduction

Time

PCA

var/va

r


(o) 3d corrections

The EURUSD rate is the best PCA

Dimension reduction of more than 2 for 2d and around 3 for 3dcorrections


Variance reduction Expected Positive Exposure - III

The distribution of the future value is more slim when a controlvariate is used

−200 −150 −100 −50 0 50 1000

0.005

0.01

0.015

0.02

0.025


V(T)

PDF

No CVCV: EURUSD PCACV: EURGBP PCACV: EURJPY PCA

(p) 2d corrections

−200 −150 −100 −50 0 50 1000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04


V(T)PDF

No CVCV: EURUSD PCACV: EURGBP PCACV: EURJPY PCA

(q) 3d corrections

again, the 3d corrections significantly improve the variance reduction


2d vs 3d corrections

Pros:

Accuracy improves significant

You can correct for all ”secondary” risk factors (stoch vol and rates)

Variance reductions get slightly better

Cons:

3d PDEs are computationally heavy

Gained variance reduction might not be worth it


Summarize

Risk measures:

Many driving risk factors (Portfolios and smile effects)

Tails are difficult (EPE, ENE and quantiles)

Foward Kolmogorov:

Deterministic modeling of the PDF

Dimension reduction:

Approximate high dimensional problems by lower dimensionalapproximations

Find the best component by looking at variance reductions of controlvariates

Method is extremely scalable (no curse of dimensionality)


References I

de Graaf, C. S. L., B. D. Kandhai, and P.M.A. Sloot.Efficient Estimation of Sensitivities for Counterparty Credit Risk withthe Finite Difference Monte-Carlo Method.Journal of Computational Finance. Forthcoming.

in ’t Hout, K. J. and S. Foulon.ADI Finite Difference Schemes For Option Pricing.International Journal of Numerical Analysis and Modeling 7(2),303–320.

Ng, L. and D. Peterson.Potential future exposure calculations using the BGM model.Wilmott Journal 1(4), 213–225.


References II

Ng, L., D. Peterson, and A. E. Rodriguez.Potential future exposure calculations of multi-asset exotic productsusing the stochastic mesh method.Journal of Computational Finance 14(2).

Itkin, A.High-Order Splitting Methods for Forward PDEs and PIDEs.International Journal of Theoretical and Applied Finance, 18(5).

Simaitis, S., C. S. L. de Graaf, N. Hari and B. D. Kandhai.Smile and Default: The Role of Stochastic Volatility and InterestRates in Counterparty Credit Risk.Submitted, June 2015.


References III

Reisinger, C. and R. Wissman.Numerical Valuation of Derivatives in High-Dimensional Settings viaPDE Expansions.Journal of Computational Finance, 18(4), 2015.

Reisinger, C. and R. Wissman.Error Analysis of Truncated Expansion Solutions to High-Dimensionalparabolic PDEs.Submitted, May 2015.


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